415 research outputs found
A moment-matching Ferguson and Klass algorithm
Completely random measures (CRM) represent the key building block of a wide
variety of popular stochastic models and play a pivotal role in modern Bayesian
Nonparametrics. A popular representation of CRMs as a random series with
decreasing jumps is due to Ferguson and Klass (1972). This can immediately be
turned into an algorithm for sampling realizations of CRMs or more elaborate
models involving transformed CRMs. However, concrete implementation requires to
truncate the random series at some threshold resulting in an approximation
error. The goal of this paper is to quantify the quality of the approximation
by a moment-matching criterion, which consists in evaluating a measure of
discrepancy between actual moments and moments based on the simulation output.
Seen as a function of the truncation level, the methodology can be used to
determine the truncation level needed to reach a certain level of precision.
The resulting moment-matching \FK algorithm is then implemented and illustrated
on several popular Bayesian nonparametric models.Comment: 24 pages, 6 figures, 5 table
Truncated Random Measures
Completely random measures (CRMs) and their normalizations are a rich source
of Bayesian nonparametric priors. Examples include the beta, gamma, and
Dirichlet processes. In this paper we detail two major classes of sequential
CRM representations---series representations and superposition
representations---within which we organize both novel and existing sequential
representations that can be used for simulation and posterior inference. These
two classes and their constituent representations subsume existing ones that
have previously been developed in an ad hoc manner for specific processes.
Since a complete infinite-dimensional CRM cannot be used explicitly for
computation, sequential representations are often truncated for tractability.
We provide truncation error analyses for each type of sequential
representation, as well as their normalized versions, thereby generalizing and
improving upon existing truncation error bounds in the literature. We analyze
the computational complexity of the sequential representations, which in
conjunction with our error bounds allows us to directly compare representations
and discuss their relative efficiency. We include numerous applications of our
theoretical results to commonly-used (normalized) CRMs, demonstrating that our
results enable a straightforward representation and analysis of CRMs that has
not previously been available in a Bayesian nonparametric context.Comment: To appear in Bernoulli; 58 pages, 3 figure
Generalized Negative Binomial Processes and the Representation of Cluster Structures
The paper introduces the concept of a cluster structure to define a joint
distribution of the sample size and its exchangeable random partitions. The
cluster structure allows the probability distribution of the random partitions
of a subset of the sample to be dependent on the sample size, a feature not
presented in a partition structure. A generalized negative binomial process
count-mixture model is proposed to generate a cluster structure, where in the
prior the number of clusters is finite and Poisson distributed and the cluster
sizes follow a truncated negative binomial distribution. The number and sizes
of clusters can be controlled to exhibit distinct asymptotic behaviors. Unique
model properties are illustrated with example clustering results using a
generalized Polya urn sampling scheme. The paper provides new methods to
generate exchangeable random partitions and to control both the cluster-number
and cluster-size distributions.Comment: 30 pages, 8 figure
Models beyond the Dirichlet process
Bayesian nonparametric inference is a relatively young area of research and it has recently undergone a strong development. Most of its success can be explained by the considerable degree of
exibility it ensures in statistical modelling, if compared to parametric alternatives, and by the emergence of new and ecient simulation techniques that make nonparametric models amenable to concrete use in a number of applied statistical problems. Since its introduction in 1973 by T.S. Ferguson, the Dirichlet process has emerged as a cornerstone in Bayesian nonparametrics. Nonetheless, in some cases of interest for statistical applications the Dirichlet process is not an adequate prior choice and alternative nonparametric models need to be devised. In this paper we provide a review of Bayesian nonparametric models that go beyond the Dirichlet process.
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